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4men-clear-a-farm-for-8-days-and-are-paid-24-How-long-will-6-men-take-to-clear-the-same-farm-if-they-are-paid-360-

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Question Number 162951 by Ar Brandon last updated on 02/Jan/22
4men clear a farm for 8 days and are paid 24$ How long will 6 men take to clear the same farm if they are paid 360$ ?
$$\mathrm{4men}\:\mathrm{clear}\:\mathrm{a}\:\mathrm{farm}\:\mathrm{for}\:\mathrm{8}\:\mathrm{days}\:\mathrm{and}\:\mathrm{are}\:\mathrm{paid}\:\mathrm{24\$} \\ $$$$\mathrm{How}\:\mathrm{long}\:\mathrm{will}\:\mathrm{6}\:\mathrm{men}\:\mathrm{take}\:\mathrm{to}\:\mathrm{clear}\:\mathrm{the}\:\mathrm{same}\:\mathrm{farm} \\ $$$$\mathrm{if}\:\mathrm{they}\:\mathrm{are}\:\mathrm{paid}\:\mathrm{360\$}\:? \\ $$
Commented by mr W last updated on 03/Jan/22
4 men earn in 8 days 24$. ⇒one man in one day: ((24)/(4×8)) 6 men earn in x days 360$. ⇒one man in one day: ((360)/(6x)) since it′s a constant how much one man earns in one day (we must assume, otherwise not solvable), ((24)/(4×8))=((360)/(6x)) ⇒x=80 days
$$\mathrm{4}\:{men}\:{earn}\:{in}\:\mathrm{8}\:{days}\:\mathrm{24\$}.\:\Rightarrow{one}\:{man}\:{in}\:{one}\:{day}:\:\frac{\mathrm{24}}{\mathrm{4}×\mathrm{8}} \\ $$$$\mathrm{6}\:{men}\:{earn}\:{in}\:{x}\:{days}\:\mathrm{360\$}.\:\Rightarrow{one}\:{man}\:{in}\:{one}\:{day}:\:\frac{\mathrm{360}}{\mathrm{6}{x}} \\ $$$${since}\:{it}'{s}\:{a}\:{constant}\:{how}\:{much}\:{one}\:{man} \\ $$$${earns}\:{in}\:{one}\:{day}\:\left({we}\:{must}\:{assume},\right. \\ $$$$\left.{otherwise}\:{not}\:{solvable}\right), \\ $$$$\frac{\mathrm{24}}{\mathrm{4}×\mathrm{8}}=\frac{\mathrm{360}}{\mathrm{6}{x}} \\ $$$$\Rightarrow{x}=\mathrm{80}\:{days} \\ $$
Commented by Ar Brandon last updated on 03/Jan/22
Got it now. Thanks Sir.
Answered by Rasheed.Sindhi last updated on 02/Jan/22
determinant (((men),(days),$),(4,8,(24)),(6,x,(360))) x:8:: { ((4:6)),((360:24)) :} x×6×24=8×4×360 x=((8×4×360)/(6×24))=80 days
$$\begin{array}{|c|c|c|}{\mathrm{men}}&\hline{\mathrm{days}}&\hline{\$}\\{\mathrm{4}}&\hline{\mathrm{8}}&\hline{\mathrm{24}}\\{\mathrm{6}}&\hline{{x}}&\hline{\mathrm{360}}\\\hline\end{array} \\ $$$${x}:\mathrm{8}::\begin{cases}{\mathrm{4}:\mathrm{6}}\\{\mathrm{360}:\mathrm{24}}\end{cases} \\ $$$${x}×\mathrm{6}×\mathrm{24}=\mathrm{8}×\mathrm{4}×\mathrm{360} \\ $$$${x}=\frac{\mathrm{8}×\mathrm{4}×\mathrm{360}}{\mathrm{6}×\mathrm{24}}=\mathrm{80}\:\mathrm{days} \\ $$
Commented by Ar Brandon last updated on 02/Jan/22
Really?! OK Sir. Thanks
Commented by Ar Brandon last updated on 02/Jan/22
Is this some sort of cross-multiplication?
Commented by Rasheed.Sindhi last updated on 03/Jan/22
The problem is somewhat strange. It is not dependant on amount of work! determinant (((men),(days),$),((4_6 ↓),(8_x ↑),(24_(360) ↑)),((men increase_(→days decrease_((inverse proportion)) ) ),,(dollars increase_(→days increase_((direct proportion)) ) )),((4:6),(x:8),(360:24)),((x:8::4:6_(A) ),,(x:8::360:24_(B) ))) A∪B: x:8:: { ((4:6)),((360:24)) :} x×6×24=8×4×360 x=((8×4×360)/(6×24))=80 days
$$\mathcal{T}{he}\:{problem}\:{is}\:{somewhat}\:{strange}. \\ $$$${It}\:{is}\:{not}\:{dependant}\:{on}\:{amount}\:{of} \\ $$$${work}! \\ $$$$\begin{array}{|c|c|c|c|c|}{\mathrm{men}}&\hline{\mathrm{days}}&\hline{\$}\\{\underset{\mathrm{6}} {\mathrm{4}}\:\:\:\downarrow}&\hline{\underset{{x}} {\mathrm{8}}\:\:\uparrow}&\hline{\underset{\mathrm{360}} {\mathrm{24}}\:\:\uparrow}\\{\underset{\underset{\left({inverse}\:{proportion}\right)} {\rightarrow{days}\:\boldsymbol{{decrease}}}} {{men}\:\boldsymbol{{increase}}}}&\hline{}&\hline{\underset{\underset{\left({direct}\:{proportion}\right)} {\rightarrow{days}\:\boldsymbol{{increase}}}} {{dollars}\:\boldsymbol{{increase}}}}\\{\mathrm{4}:\mathrm{6}}&\hline{{x}:\mathrm{8}}&\hline{\mathrm{360}:\mathrm{24}}\\{\underset{\mathrm{A}} {\underbrace{{x}:\mathrm{8}::\mathrm{4}:\mathrm{6}}}}&\hline{}&\hline{\underset{{B}} {\underbrace{{x}:\mathrm{8}::\mathrm{360}:\mathrm{24}}}}\\\hline\end{array} \\ $$$$\mathrm{A}\cup\mathrm{B}: \\ $$$${x}:\mathrm{8}::\begin{cases}{\mathrm{4}:\mathrm{6}}\\{\mathrm{360}:\mathrm{24}}\end{cases} \\ $$$${x}×\mathrm{6}×\mathrm{24}=\mathrm{8}×\mathrm{4}×\mathrm{360} \\ $$$${x}=\frac{\mathrm{8}×\mathrm{4}×\mathrm{360}}{\mathrm{6}×\mathrm{24}}=\mathrm{80}\:\mathrm{days} \\ $$
Commented by Rasheed.Sindhi last updated on 03/Jan/22
This is by ′proportion method′ while sir mr W ′s method (comment to the question) is ′unitary method′ This is a compound proportion.
$$\mathcal{T}{his}\:{is}\:{by}\:'{proportion}\:{method}'\:{while} \\ $$$${sir}\:{mr}\:{W}\:'{s}\:{method}\:\left({comment}\:{to}\:\right. \\ $$$$\left.{the}\:{question}\right)\:{is}\:'{unitary}\:{method}' \\ $$$$\mathcal{T}{his}\:{is}\:{a}\:{compound}\:{proportion}. \\ $$
Commented by Rasheed.Sindhi last updated on 03/Jan/22
Sir,pl read my above comment again as I′ve edited it before receiving your comment.
$$\boldsymbol{{Sir}},{pl}\:{read}\:{my}\:{above}\:{comment}\:{again} \\ $$$${as}\:{I}'{ve}\:{edited}\:{it}\:{before}\:{receiving}\:{your} \\ $$$${comment}. \\ $$
Commented by Rasheed.Sindhi last updated on 03/Jan/22
I have also sampathy for them:) But sir, the question says: 4 men.....∣6men......clear same farm. that is do same quantity of work.
$$\left.{I}\:{have}\:{also}\:{sampathy}\:{for}\:{them}:\right) \\ $$$${But}\:\boldsymbol{{sir}},\:{the}\:{question}\:{says}: \\ $$$$\mathrm{4}\:{men}…..\mid\mathrm{6}{men}……{clear}\:\boldsymbol{{same}}\:\boldsymbol{{farm}}. \\ $$$${that}\:{is}\:{do}\:\boldsymbol{{same}}\:\boldsymbol{{quantity}}\:\boldsymbol{{of}}\:\boldsymbol{{work}}. \\ $$
Commented by mr W last updated on 03/Jan/22
“clear the same farm” doesn′t define the quantity of the work, only the type of the work, i think. it could be that the 4 men cleared the first 120 m^2 of the farm in 8 days for totally 24$, and the 6 men cleared the remaining 1800 m^2 of the same farm in 80 days for totally 360$. the same farm pays the same money for each man and each day. different farms may pay less or more. therefore it makes sinse to state that they cleared the same farm. this is my opinion.
$$“{clear}\:{the}\:{same}\:{farm}''\:{doesn}'{t}\:{define} \\ $$$${the}\:{quantity}\:{of}\:{the}\:{work},\:{only}\:{the} \\ $$$${type}\:{of}\:{the}\:{work},\:{i}\:{think}.\:\:{it}\:{could}\:{be} \\ $$$${that}\:{the}\:\mathrm{4}\:{men}\:{cleared}\:{the}\:{first}\:\mathrm{120}\:{m}^{\mathrm{2}} \\ $$$${of}\:{the}\:{farm}\:{in}\:\mathrm{8}\:{days}\:{for}\:{totally}\:\mathrm{24\$}, \\ $$$${and}\:{the}\:\mathrm{6}\:{men}\:{cleared}\:{the}\:{remaining} \\ $$$$\mathrm{1800}\:{m}^{\mathrm{2}} \:{of}\:{the}\:{same}\:{farm}\:{in}\:\mathrm{80}\:{days} \\ $$$${for}\:{totally}\:\mathrm{360\$}.\:{the}\:{same}\:{farm}\:{pays} \\ $$$${the}\:{same}\:{money}\:{for}\:{each}\:{man}\:{and} \\ $$$${each}\:{day}.\:{different}\:{farms}\:{may}\:{pay} \\ $$$${less}\:{or}\:{more}.\:{therefore}\:{it}\:{makes} \\ $$$${sinse}\:{to}\:{state}\:{that}\:{they}\:{cleared}\:{the} \\ $$$${same}\:{farm}.\:{this}\:{is}\:{my}\:{opinion}. \\ $$
Commented by mr W last updated on 03/Jan/22
but no matter how it is, 4 men earn only 24$ in 8 days, that′s definitely too less, i think :)
$${but}\:{no}\:{matter}\:{how}\:{it}\:{is},\:\mathrm{4}\:{men}\:{earn} \\ $$$${only}\:\mathrm{24\$}\:{in}\:\mathrm{8}\:{days},\:{that}'{s}\:{definitely} \\ $$$$\left.{too}\:{less},\:{i}\:{think}\::\right) \\ $$
Commented by Rasheed.Sindhi last updated on 03/Jan/22
Okay sir, you′re keen obserer,no doubt! “4 men clear a farm...”means they clear whole farm.“...6 men take to clear the same farm”,doesn′t again it mean the whole farm?
$$\mathrm{Okay}\:\mathrm{sir},\:{you}'{re}\:{keen}\:{obserer},{no}\:{doubt}! \\ $$$$“\mathrm{4}\:\mathrm{men}\:\mathrm{clear}\:\mathrm{a}\:\mathrm{farm}…''{means}\:{they} \\ $$$${clear}\:\boldsymbol{{whole}}\:{farm}.“…\mathrm{6}\:\mathrm{men}\:\mathrm{take} \\ $$$$\mathrm{to}\:\mathrm{clear}\:\mathrm{the}\:\mathrm{same}\:\mathrm{farm}'',{doesn}'{t}\:\:{again}\:{it}\: \\ $$$${mean}\:{the}\:{whole}\:{farm}? \\ $$
Commented by mr W last updated on 03/Jan/22
actually i thought the same as you. but then i got a problem with the logic. so i just changed my opinion in order to become conform with the solution.
$${actually}\:{i}\:{thought}\:{the}\:{same}\:{as}\:{you}. \\ $$$${but}\:{then}\:{i}\:{got}\:{a}\:{problem}\:{with}\:{the}\:{logic}. \\ $$$${so}\:{i}\:{just}\:{changed}\:{my}\:{opinion}\:{in}\:{order} \\ $$$${to}\:{become}\:{conform}\:{with}\:{the}\:{solution}. \\ $$
Commented by mr W last updated on 03/Jan/22
this is the logic problem i mentioned. “clear a farm” and “clear the same farm” may not mean the same work done, because one can not pay for the same work done in one time only 24$ and in an other time 360$. certainly in reality it′s possible, but in mathematics it′s not possible. because there is no mathematics if there is no ligic. therefore “clear a farm” and “clear the same farm” can logically only mean the type of the work, i.e. “clear in a farm” as well as “clear in the same farm”. i think the guy who originally wrote this question is no lawyer, even no native speaker of english, like you and me. therefore we should look more at the logic behind it instead of the wording.
$${this}\:{is}\:{the}\:{logic}\:{problem}\:{i}\:{mentioned}. \\ $$$$“{clear}\:{a}\:{farm}''\:{and}\:“{clear}\:{the}\:{same} \\ $$$${farm}''\:{may}\:{not}\:{mean}\:{the}\:{same}\:{work} \\ $$$${done},\:{because}\:{one}\:{can}\:{not}\:{pay}\:{for}\:{the} \\ $$$${same}\:{work}\:{done}\:{in}\:{one}\:{time}\:{only}\:\mathrm{24\$}\: \\ $$$${and}\:{in}\:{an}\:{other}\:{time}\:\mathrm{360\$}.\:{certainly}\: \\ $$$${in}\:{reality}\:{it}'{s}\:{possible},\:{but}\:{in}\: \\ $$$${mathematics}\:{it}'{s}\:{not}\:{possible}. \\ $$$${because}\:{there}\:{is}\:{no}\:{mathematics}\:{if} \\ $$$${there}\:{is}\:{no}\:{ligic}.\:{therefore}\:“{clear}\:{a}\: \\ $$$${farm}''\:{and}\:“{clear}\:{the}\:{same}\:{farm}'' \\ $$$${can}\:{logically}\:{only}\:{mean}\:{the}\:{type}\:{of} \\ $$$${the}\:{work},\:{i}.{e}.\:“{clear}\:{in}\:{a}\:{farm}''\:{as}\: \\ $$$${well}\:{as}\:“{clear}\:{in}\:{the}\:{same}\:{farm}''. \\ $$$${i}\:{think}\:{the}\:{guy}\:{who}\:{originally}\:{wrote}\: \\ $$$${this}\:{question}\:{is}\:{no}\:{lawyer},\:{even}\:{no} \\ $$$${native}\:{speaker}\:{of}\:{english},\:\:{like}\:{you} \\ $$$${and}\:{me}.\:{therefore}\:{we}\:{should}\:{look}\:{more} \\ $$$${at}\:{the}\:{logic}\:{behind}\:{it}\:{instead}\:{of}\:{the} \\ $$$${wording}. \\ $$
Commented by Rasheed.Sindhi last updated on 03/Jan/22
T_(H_(AN) K) S S_I_(R!) 👍👍👍👍👍
$$\mathbb{T}_{\mathbb{H}_{\mathbb{AN}} \mathbb{K}} \mathbb{S}\:\mathbb{S}_{\mathbb{I}_{\mathbb{R}!} } \\ $$👍👍👍👍👍

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